Ramifications, Reduction of Singularities, Separatrices

Abstract

We study the behaviour of sequences of blowing-ups under ramifications, and use the results to give a simple proof of Camacho-Sad's Theorem on the existence of Separatrices for singularities of plane holomorphic foliations. The main result we prove is that for any finite sequence $π$ of blowing-ups, there is a ramification morphism $ρ$ such that the elimination of indeterminations $\tildeπ$ of $π^{-1}\circ ρ$ is a sequence of blowing-ups \emph{with centers at regular points of the exceptional divisors}; moreover, we show that if $π$ is the reduction of singularities of a foliation $\mathcal{F}$, then $ρ$ can be such that $\tildeπ^{\star}ρ^{\star}\mathcal{F}$ has only simple singularities.